Detailed study of quark-hadron duality in spin structure functions of the proton and neutron

V. Lagerquist, S. E. Kuhn, and N. Sato

Phys. Rev. C 107, 045201 – Published 6 April 2023

Abstract

Background: The response of hadrons, the bound states of the strong force (QCD), to external probes can be described in two different, complementary frameworks: as direct interactions with their fundamental constituents, quarks and gluons, or alternatively as elastic or inelastic coherent scattering that leaves the hadrons in their ground state or in one of their excited (resonance) states. The former picture emerges most clearly in hard processes with high momentum transfer, where the hadron response can be described by the perturbative expansion of QCD, while at lower energy and momentum transfers, the resonant excitations of the hadrons dominate the cross section. The overlap region between these two pictures, where both yield similar predictions, is referred to as quark-hadron duality and has been extensively studied in reactions involving unpolarized hadrons. Some limited information on this phenomenon also exists for polarized protons, deuterons, and ${}^{3}{He}$ nuclei, but not yet for neutrons.

Purpose: In this paper, we present comprehensive and detailed results on the correspondence between the extrapolated deep inelastic structure function $g_{1}$ of both the proton and the neutron with the same quantity measured in the nucleon resonance region. Thanks to the fine binning and high precision of our data, and using a well-controlled perturbative QCD (pQCD) fit for the partonic prediction, we can make quantitative statements about the kinematic range of applicability of both local duality and global duality.

Method: We use the most updated QCD global analysis results at high $x$ from the Jefferson Lab Angular Momentum Collaboration to extrapolate the spin structure function $g_{1}$ into the nucleon resonance region and then integrate over various intervals in the scaling variable $x$ . We compare the results with the large data set collected in the quark-hadron transition region by the CLAS Collaboration, including, for the first time, deconvoluted neutron data, integrated over the same intervals. We present this comparison as a function of the momentum transfer $Q^{2}$ .

Results: We find that, depending on the integration interval and the minimum momentum transfer chosen, a clear transition to quark-hadron duality can be observed in both nucleon species. Furthermore, we show, for the first time, the approach to scaling behavior for $g_{1}$ measured in the resonance region at sufficiently high momentum transfer.

Conclusions: Our results can be used to quantify the deviations from the applicability of pQCD for data taken at moderate energies and can help with extraction of quark distribution functions from such data.

Received 6 May 2022
Revised 21 November 2022
Accepted 20 March 2023

DOI:https://doi.org/10.1103/PhysRevC.107.045201

Physics Subject Headings (PhySH)

Particle interactions Quantum chromodynamics Quark model

Baryons

Spin

Nuclear PhysicsParticles & Fields

Authors & Affiliations

V. Lagerquist ¹, S. E. Kuhn ^1,*, and N. Sato²

¹Old Dominion University, Norfolk, Virginia 23529, USA
²Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA

^*Corresponding author: skuhn@odu.edu

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Vol. 107, Iss. 4 — April 2023

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Images

Figure 1
Feynman diagram for inclusive electron scattering off a nucleon target. $W$ is the invariant mass of the unobserved final-state $X$ . All other symbols are explained in the text.
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Figure 2
Schematic dependence of the measured structure function $F_{2}$ in inelastic electron scattering off the nucleon on the variable $ω^{'} = W^{2} / Q^{2} + 1$ , which is close to $1 / x$ at large $Q^{2}$ . Panels (a) through (d) are for increasing four-momentum transfer $Q^{2}$ . As can be observed, the resonance excitations of the nucleon are most prominent at low $Q^{2}$ , while at higher $Q^{2}$ the curve for $F_{2}$ approaches the scaling limit (dashed line), hence indicating a transition to quark-hadron duality in this observable. Reproduced from the paper by Bloom and Gilman [3], with the permission of AIP Publishing.
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Figure 3
Representation of the experimental data set used in this analysis. The measured data points are binned in bins in $Q^{2}$ , as indicated by the shaded area for the example of the bin $1.87 {GeV}^{2} < Q^{2} < 2.23 {GeV}^{2}$ ; see also Table 2. The truncated integrals are then formed over specific regions in $W$ as spelled out in Table 1.
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Figure 4
Test of duality for truncated integrals. We show integrals ${\bar{Γ}}_{1} (Δ W, Q^{2})$ of the spin structure function $g_{1} (x, Q^{2})$ over regions in $x$ corresponding to six fixed regions $Δ W$ of the final-state mass $W$ , plotted as a function of $Q^{2}$ . (a) The region of the first excited state of the nucleon, the $Δ (1232)$ resonance. (b) The region of the $N (1440) 1 / 2^{+}, N (1520) 3 / 2^{-}$ , and $N (1535) 1 / 2^{-}$ resonances. (c) The region including the $N (1680) 5 / 2^{+}$ resonance. (d) The remainder of the customary resonance region, 1.82 GeV $< W <$ 2 GeV. (e) The sum of regions in panels (a) through (d), i.e., the entire resonance region 1.07 GeV $< W <$ 2 GeV. (f) Same as panel (e), with the elastic peak included: 0.938 GeV $< W < 2$ GeV (corresponding to a range in $x$ extending all the way to $x = 1$ ). The top (red) bands and data points (circles) are for the proton, and the bottom (blue) bands and data points (triangles) are for the neutron. The data points are shown with statistical and systematic uncertainties added in quadrature (error bars). The solid bands show the full prediction from the extrapolated JAM fit, including target mass and higher-twist contributions. The striped band (proton) and the cross-hatched band (neutron) show the results including only the leading-twist contribution.
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Figure 5
Approach towards the scaling limit for the structure function $g_{1} (x, Q^{2})$ , averaged over 11 different $x$ bins of width $Δ x = 0.05$ , as a function of $Q^{2}$ . Data and JAM bands are shown multiplied with the average $x$ for each bin for better clarity; symbols are the same as those in Fig. 4. The vertical dashed line indicates the limit $W = 2$ GeV of the resonance region, which lies to the left. Last panel: Kinematic location of all data points from EG1b in the $x$ vs $Q^{2}$ plane [red, proton; blue, neutron). The gray band indicates a sample interval in $x$ over which the data are averaged [corresponding to panel (a)], and the vertical lines indicate the nominal central values of each $Q^{2}$ bin.
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